shaders - Rotate mesh to normal

I have some instanced geometry (basic tube meshes) laid out in a grid, and I have a noise texture (normal map) that I want to use to rotate my instances with. So head pixel in my texture is a normal and I want to rotate each instance with the corresponding normal in a shader. How can I achieve that in a shader only? Unless I am mistaking, there should only be a rotation around the X and Z axes.

Answer

You can get a rotation matrix from a a direction ($\vec d$) and an up vector ($\vec u$) (direction is the normal here).

First you need to get 2 perpendicular vectors, one pointing to the right ($\vec r$) and one pointing up ($\vec t$) when looking in the direction of the normal vector.

You can get $\vec r$ by getting the cross product of $\vec d$ and $\vec u$:

$$\vec r = \vec d \times \vec u$$

Then you can get $\vec t$ by doing the same to $\vec r$ and $\vec d$:

$$\vec t = \vec r \times \vec d$$

The rotation matrix can be defined by these three vectors in the following way:

$$\begin{bmatrix} \vec r_x & \vec r_y & \vec r_z & 0 \\ \vec t_x & \vec t_y & \vec t_z & 0 \\ \vec d_x & \vec d_y & \vec d_z & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$$

Use it as a model matrix